Universal Features of Chiral Symmetry Breaking in Large-$N$ QCD

TL;DR

The paper tests the universality of chiral symmetry breaking in large-$N$ QCD by comparing non-perturbative low-lying Dirac spectra from a chiral TEK lattice with chiral Random Matrix Theory predictions. It uses twisted-volume reduction to reach up to $N=841$ and implements overlap fermions via a truncated overlap to preserve chiral symmetry, enabling scale-invariant and parameter-dependent RMT comparisons. The results show that the low-lying spectrum adheres to RMT in the large-$N$ (epsilon) regime and yield a consistent large-$N chiral condensate $\Sigma/N$, renormalized to $\Sigma_{\mathrm R}/(N\sqrt{\sigma^3}) \approx 0.080(6)$, in agreement with Wilson-TEK determinations within uncertainties and with the expectation of smaller lattice artifacts for overlap quarks. Overall, the study validates the universal RMT description of chiral symmetry breaking in large-$N$ QCD and demonstrates the TEK approach as a viable route to explore this regime, laying groundwork for continuum extrapolations and extensions to other large-$N$ gauge theories.

Abstract

We investigate the universal features of chiral symmetry breaking in large-$N$ QCD by comparing non-perturbative determinations of the low-lying Dirac spectrum with chiral Random Matrix Theory (RMT) predictions. Our numerical Monte Carlo calculations are based on a chiral lattice discretization of the Dirac operator, and exploit twisted volume reduction to reach $N$ as large as 841. Matching lattice data with RMT analytic results, we are able to extract the large-$N$ chiral condensate, which is compared with a recent determination obtained with non-chiral Wilson quarks from twisted volume-reduced models.

Figures (7)

• Figure 1: First two complex eigenvalues of the truncated overlap Dirac operator obtained with $M=1.2$, $N_5=24$ and stout-smeared gauge links with stout parameters $\rho=0.1$, $n_s=5$, corresponding to a smoothing radius of two lattice spacings. The solid black line stands for the expected overlap circle \ref{['eq:overlap_circle']} in the complex plane. Left panel: $N=529$, $b=0.355$. Right panel: $N=841$, $b=0.360$.
• Figure 2: Ratios of expectation values $\braket{\lambda_{k_1}}/\braket{\lambda_{k_2}}$, obtained considering all combinations with $k_i,k_j\le 4$, divided by the same quantity computed in the RMT model, $\braket{z_{k_1}}_{{{\rm RMT}}}/\braket{z_{k_2}}_{{{\rm RMT}}}$. The red dashed line at $y=1$ represents the agreement between lattice data and RMT predictions. Top and bottom panels refer, respectively, to $b=0.360$ and $b=0.355$.
• Figure 3: Distribution of the ratio of the first two Dirac eigenvalues, $r=\lambda_{1}/\lambda_{2}$. Top panel: $N=529$, $b=0.355$. Bottom panel: $N=841$, $b=0.360$. The dashed curve represents the parameter-free functional form predicted by RMT for $p(r)$, reported in Eq. \ref{['eq:p_r_prediction']}. In all plots we also show the constant bin size $\Delta r$ used to obtain the displayed distributions.
• Figure 4: Best fits of the probability distribution of the first (top panels) and the second (bottom panels) eigenvalues according to, respectively Eqs. \ref{['eq:p_1_prediction']} and \ref{['eq:p_2_prediction']}. Left panels refer to $N=529,b=0.355$, right panels to $N=841,b=0.360$. In all plots we also show the constant bin sized $\Delta x_1$ and $\Delta x_2$ used to obtain the displayed distributions.
• Figure 5: The bare large-$N$ chiral condensate $\Sigma/N$ in lattice units as a function of $1/\sqrt{V}$. Left panel: shown curves are polynomial fits in $1/\sqrt{V}$ to each individual data set. Right panel: shown curves are the result of a global best fit according to the empirical ansatz in Eq. \ref{['eq:bestfit_cc_vs_N']} (bottom panel).